The numbersxnsubscriptxnx_{n} are all below a finiteboundMMM. For demonstrating this, we write the inequalityxn<xn+1subscriptxnsubscriptxn1x_{n}<x_{{n+1}} in the form xn<a+xnqsubscriptxnqasubscriptxnx_{n}<\sqrt[q]{a+x_{n}}, which implies xnq<a+xnsuperscriptsubscriptxnqasubscriptxnx_{n}^{q}<a+x_{n}, i.e.

Taking limits of both sides of (1) we see that x′=a+x′qsuperscriptxnormal-′qasuperscriptxnormal-′x^{{\prime}}=\sqrt[q]{a+x^{{\prime}}}, i.e. x′⁣q-x′-a=0superscriptxnormal-′qsuperscriptxnormal-′a0x^{{\prime q}}-x^{{\prime}}-a=0, which means that x′=Msuperscriptxnormal-′Mx^{{\prime}}=M, in other words: the limit of the sequence is the only positive rootMMM of the equation (3).